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{SECT 0 {PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 256 
45 "This file contains some Maple animations for " }}{PARA 264 "" 0 "
" {TEXT 257 16 "GENERAL SURFACES" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 263 78 "Depending on the hardware you are using,
 you may want to decrease/increase the" }}{PARA 0 "" 0 "" {TEXT 264 
35 "number of frames in the animations." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 55 "We animate first the Moebius
 strip in various settings:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "mob1:=
animate(\n[0.05*cos(y)+2*cos(t),0.05*sin(y)+2*sin(t),\ny=0..2*Pi],t=0.
.2*Pi,color=red,frames=20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
82 "mob2:=animate(\n[cos(t/2)*x+5,sin(t/2)*x,x=-2..2],t=0..2*Pi,\ncolo
r=blue,frames=20):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 258 69 "The follo
wing animation shows simultaneously the motion of the center" }}{PARA 
0 "" 0 "" {TEXT 259 71 "of a segment in a plane and the motion of the \+
segment in an orthogonal " }}{PARA 0 "" 0 "" {TEXT 260 7 "plane. " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "display(\{mob1,mob2\},thick
ness=3,axes=none,\nscaling=constrained,\ntitlefont=[COURIER,BOLD,18],
\ntitle=`Animating the Moebius strip in 4d`);" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT 261 77 "The next animation is in 3d; it shows the rotating \+
segment while its midpoint" }}{PARA 0 "" 0 "" {TEXT 262 24 "revolves a
long a circle." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "mob3:=anim
ate3d([cos(t)*y,sin(t)*y,0],\nx=0..2,y=0..2,t=0..2*Pi,\ncolor=blue,fra
mes=20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "mob4:=animate3
d(\n[cos(t)*(2+(x-1)*cos(t/2)),\nsin(t)*(2+(x-1)*cos(t/2)),(x-1)*sin(t
/2)],\nx=0..2,y=0..2,t=0..2*Pi,\ncolor=red,frames=20):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "display3d(\{mob3,mob4\},\nscaling=
constrained,axes=normal,\ntickmarks=[0,0,0],orientation=[113,74],\nthi
ckness=3,\ntitlefont=[COURIER,BOLD,18],\ntitle=`Animating the Moebius \+
strip in 3d`);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 265 52 "We now see t
hat the segment sweeps the Moebius band." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 264 "animate3d([cos(t*u)*(2+v*cos(t*u/2)),\nsin(t*u)*(2+v
*cos(t*u/2)),v*sin(t*u/2)],\nu=0..2*Pi,v=-1..1,t=0..1,\nthickness=3,co
lor=v,style=patchnogrid,\nscaling=constrained,axes=none,frames=20,\nti
tlefont=[COURIER,BOLD,18],\ntitle=`Animating the finite Moebius Strip \+
in 3d`);" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 35 "A construction of
 the Klein Bottle:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "kb1:=p
lot3d([2*sin(t),2*cos(t),u],\nt=Pi/2..5*Pi/2,u=-4..0,\ngrid=[20,10],st
yle=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "kb2:=
plot3d([(4+2*cos(Pi*u/14+4*Pi/14))*sin(t),\n(4+2*cos(Pi*u/14+4*Pi/14))
*cos(t),u],\nt=Pi/2..5*Pi/2,u=-4..10,\nstyle=wireframe):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "kb3:=plot3d([cos(t)*(5+2*cos(u))+5
,2*sin(u),\nsin(t)*(5+2*cos(u))],\nt=Pi/2..Pi,u=Pi..3*Pi,style=patchno
grid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "kb4:=plot3d([cos
(t)*(5+2*cos(u))+5,2*sin(u),\nsin(t)*(5+2*cos(u))+10],t=-Pi/2..Pi,u=Pi
..3*Pi,\nstyle=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 108 "kb5:=plot3d([cos(t)*(4+2*cos(u)),\nsin(t)*(4+2*cos(u)),2*sin(u)
-4],\nt=Pi..3*Pi,u=Pi..2*Pi,style=patchnogrid):" }}}{SECT 0 {PARA 4 "
" 0 "" {TEXT 268 35 "Here is the Plumber's Klein Bottle:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "display3d(\{kb1,kb2,kb3,kb4,kb5\},
\norientation=[-27,-92],\nshading=zhue,scaling=constrained,\ntitlefont
=[COURIER,BOLD,18],\ntitle=`The Plumber's Klein Bottle`);" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "halfkb1:=plot3d([2*sin(t),2*cos(t),
u],\nt=Pi/2..3*Pi/2,u=-4..0,grid=[20,10]):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 116 "halfkb2:=plot3d([\n(4+2*cos(Pi*u/14+4*Pi/14))*sin(
t),\n(4+2*cos(Pi*u/14+4*Pi/14))*cos(t),u],\nt=Pi/2..3*Pi/2,u=-4..10):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "halfkb3:=plot3d([cos(t)
*(5+2*cos(u))+5,2*sin(u),\nsin(t)*(5+2*cos(u))],t=Pi/2..Pi,u=Pi..2*Pi)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "halfkb4:=plot3d([cos(t
)*(5+2*cos(u))+5,2*sin(u),\nsin(t)*(5+2*cos(u))+10],t=-Pi/2..Pi,u=Pi..
2*Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "halfkb5:=plot3d([
cos(t)*(4+2*cos(u)),\nsin(t)*(4+2*cos(u)),2*sin(u)-4],\nt=Pi..2*Pi,u=P
i..2*Pi):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 267 43 "Here is half of th
e Plumber's Klein Bottle:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
202 "display3d(\n\{halfkb1,halfkb2,halfkb3,halfkb4,halfkb5\},\norienta
tion=[-88,-33],\nshading=zhue,style=patchnogrid,\nscaling=constrained,
\ntitlefont=[COURIER,BOLD,18],title=`Half of the Plumber's Klein Bottl
e`);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 269 46 "Half of the Klein Bott
le vs the Moebius strip:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 
"moebiusstrip:=plot3d(\n[25+5*cos(u)*(2+v*cos(u/2)),\n-2+5*sin(u)*(2+v
*cos(u/2)),5*v*sin(u/2)],\nu=0..2*Pi,v=-1..1,\nstyle=patchnogrid,scali
ng=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "displa
y3d(\n\{halfkb1,halfkb2,halfkb3,halfkb4,halfkb5,\nmoebiusstrip\},\nori
entation=[-77,-53],shading=zhue,\ntitlefont=[COURIER,BOLD,18],\ntitle=
`Half of the Klein Bottle vs the Moebius strip`);" }}}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT 270 56 "We now create some animations about cutting-an
d-pasting:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "twistrec:=ani
mate3d(\n[(1-t)*x+t*exp(x)*cos(y),(1-t)*y+t*exp(x)*sin(y),0],x=Pi/4..P
i/2,y=0..Pi,t=0..1,\ncolor=x,grid=[15,40]):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 55 "d:=(exp(Pi/2)+exp(Pi/4))/2:\nr:=(exp(Pi/2)-exp(Pi/
4))/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "curlrec:=animate
3d(\n[(d+(4*r/Pi)*sin((Pi/(4*r))*t*(exp(x)-d))/t)*\ncos((1+t/4)*y),(d+
(4*r/Pi)*sin((Pi/(4*r))*t*\n(exp(x)-d))/t)*sin((1+t/4)*y),((4*r)/Pi)*
\n(1-cos((Pi/(4*r))*t*(exp(x)-d)))/t],\nx=Pi/4..Pi/2,y=0..Pi,t=-0.005.
.4,color=x,\ngrid=[15,40]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
212 "display([twistrec,curlrec],insequence=true,\nstyle=patchnogrid,th
ickness=2,\nscaling=constrained,orientation=[-24,64],\ntitlefont=[COUR
IER,BOLD,18],\ntitle=`The torus is a rectangle with opposite \nsides i
dentified`);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 59 "Instead of rota
ting a segment we now rotate a figure eight." }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 91 "mob5:=animate([0.05*cos(y)+2*cos(t),\n0.05*sin(y)+
2*sin(t),y=0..2*Pi],t=0..2*Pi,\ncolor=red):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 121 "mob6:=animate(\n[4+cos(t/2)*sin(v)-sin(t/2)*sin(2
*v),\nsin(t/2)*sin(v)+cos(t/2)*sin(2*v),v=0..2*Pi],\nt=0..2*Pi,color=b
lue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "mob7:=animate(\n[
4+cos(t/2)*sin(v)-sin(t/2)*sin(2*v),\nsin(t/2)*sin(v)+cos(t/2)*sin(2*v
),v=-Pi/2..Pi/2],t=0..2*Pi,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 126 "display([mob5,mob6],\nthickness=3,axes=none,scaling=
constrained,\ntitlefont=[COURIER,BOLD,18],\ntitle=`Rotating a figure e
ight`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "display([mob5,m
ob7],\nthickness=3,axes=none,\ntitlefont=[COURIER,BOLD,18],\ntitle=`Ro
tating half of a figure eight`);" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
274 25 "Animate the Klein Bottle:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 349 "animate3d(\n[(2+cos(t*u/2)*sin(v)-sin(t*u/2)*sin(2*v
))*\ncos(t*u),(2+cos(t*u/2)*sin(v)-sin(t*u/2)*\nsin(2*v))*sin(t*u),sin
(t*u/2)*sin(v)+cos(t*u/2)*\nsin(2*v)],u=0..2*Pi,v=0..2*Pi,t=0..1,\ngri
d=[30,40],orientation=[-40,53],\nscaling=constrained,style=patchnogrid
,\ncolor=sin(v/2),frames=20,\ntitlefont=[COURIER,BOLD,18],\ntitle=`The
 Klein Bottle revisited`);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 273 33 "
Animate half of the Klein Bottle:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 361 "animate3d(\n[(2+cos(t*u/2)*sin(v)-sin(t*u/2)*sin(2*v
))*\ncos(t*u),(2+cos(t*u/2)*sin(v)-sin(t*u/2)*\nsin(2*v))*sin(t*u),sin
(t*u/2)*sin(v)+cos(t*u/2)*\nsin(2*v)],u=0..2*Pi,v=-Pi/2..Pi/2,t=0..1,
\ngrid=[30,20],orientation=[-40,53],\nscaling=constrained,style=patchn
ogrid,\ncolor=sin(v/2),frames=20,\ntitlefont=[COURIER,BOLD,18],\ntitle
=`Half of the Klein Bottle revisited`);" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT 272 34 "Still picture of the Klein Bottle:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 318 "plot3d(\n[(2+cos(u/2)*sin(v)-sin(u/2)*sin(2*v
))*cos(u),\n(2+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*sin(u),\nsin(u/2)*si
n(v)+cos(u/2)*sin(2*v)],\nu=0..2*Pi,v=0..2*Pi,grid=[30,60],\norientati
on=[45,45],scaling=constrained,\nstyle=patchnogrid,color=sin(v/2),\nti
tlefont=[COURIER,BOLD,18],\ntitle=`Still picture of the Klein Bottle`)
;" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 271 42 "Still picture of half of \+
the Klein Bottle:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "plot3d
(\n[(2+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*cos(u),\n(2+cos(u/2)*sin(v)-
sin(u/2)*sin(2*v))*sin(u),\nsin(u/2)*sin(v)+cos(u/2)*sin(2*v)],\nu=0..
2*Pi,v=-Pi/2..Pi/2,\ngrid=[30,60],orientation=[45,45],\nscaling=constr
ained,style=patchnogrid,\ncolor=sin(v/2),\ntitlefont=[COURIER,BOLD,18]
,\ntitle=`Still picture of half of the Klein Bottle`);" }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT 275 46 "Mobius strip bent to half of the Klein B
ottle:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "animate3d(\n[(2+c
os(u/2)*sin(t*v+Pi)-sin(u/2)*\nsin(2*(t*v+Pi)))*cos(u),\n(2+cos(u/2)*s
in(t*v+Pi)-sin(u/2)*\nsin(2*(t*v+Pi)))*sin(u),\nsin(u/2)*sin(t*v+Pi)+c
os(u/2)*sin(2*(t*v+Pi))],u=0..2*Pi,v=-Pi/2..Pi/2,t=1/4..1,\ngrid=[50,2
5],scaling=constrained,\nstyle=patchnogrid,color=sin(v/2),\ntitlefont=
[COURIER,BOLD,18],\ntitle=`Mobius strip bent to half of the Klein Bott
le`);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 115 "The following animati
on may take a lot of time to load! It shows how two Mobius strips join
 to form a Klein Bottle." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 
"paste1:=animate3d(\n[(2+cos(u/2)*sin(t*v)-sin(u/2)*sin(2*t*v))*\ncos(
u),\n(2+cos(u/2)*sin(t*v)-sin(u/2)*sin(2*t*v))*\nsin(u),sin(u/2)*sin(t
*v)+cos(u/2)*sin(2*t*v)],\nu=0..2*Pi,v=-Pi/2..Pi/2,t=1/4..1,\nframes=2
0,grid=[30,15],color=sin(v/2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 264 "paste2:=animate3d(\n[(2+cos(u/2)*sin(t*v+Pi)-sin(u/2)*\nsin(2
*(t*v+Pi)))*cos(u),\n(2+cos(u/2)*sin(t*v+Pi)-sin(u/2)*\nsin(2*(t*v+Pi)
))*sin(u),\nsin(u/2)*sin(t*v+Pi)+cos(u/2)*sin(2*(t*v+Pi))+5-5*t],u=0..
2*Pi,v=-Pi/2..Pi/2,t=1/4..1,\nframes=20,grid=[30,15],color=sin(v/2)):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "display3d([paste1,paste
2],\nstyle=patchnogrid,scaling=constrained,\nshading=Z,orientation=[-1
04,71]);" }}}}}}{MARK "10 11 3 0 0" 66 }{VIEWOPTS 1 1 0 3 2 1804 }